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Polynomial functions are algebraic expressions that consist of terms in the form of \(a_nx^n\), where \(a_n\) is a coefficient, and \(n\) is a non-negative integer called the degree of the term. These functions are sum of a finite number of these terms with real coefficients and are generally written in descending order of degrees. For example, \(f(x) = 4x^3 - 3x^2 + 2x - 1\) is a polynomial of degree 3, as the highest power of \(x\) is 3.

The special aspect of polynomial functions is that they are continuous and smooth, which means they do not have breaks, holes, or sharp corners anywhere in their graphs. Additionally, they are defined for all real numbers, which signifies that you can input any real number, and the polynomial function will give you a corresponding output without any complications from undefined operations, like division by zero or taking the square root of a negative number.

This fundamental characteristic of polynomials simplifies the process of determining their domain — the set of all possible inputs — because the domain for a polynomial function in \(x\) is always all real numbers. The absence of any restrictions like roots or divisions in a polynomial means no need to exclude any numbers from the domain.

The real numbers encompass the entire spectrum of numbers used for measuring continuous quantities. They include the rational numbers, such as fractions and integers, as well as the irrational numbers; numbers that cannot be expressed as simple fractions and have non-repeating, non-terminating decimal expansions like \(\pi\) or \(\sqrt{2}\).

When we talk about the domain of a function in terms of real numbers, we imply that for every real number that exists, the function can accept it as an input and provide an output. In the context of polynomial functions, since they don't have restrictions such as needing a non-negative input for a square root, or avoiding a zero in the denominator of a fraction, their domain is simply all real numbers.

By stating the domain is all real numbers, we allow any number along the continuous number line to be part of the function's inputs, showcasing the truly vast and unrestricted nature of polynomials when it comes to their domain.

Set notation is a shorthand way to express collections of objects, numbers, or entities, and it's widely used in mathematics to clearly define the domain or range of a function. When we use set notation, we employ curly brackets \(\{\}\) to enclose our elements to represent a set.

For instance, the domain of a function can be expressed in set notation. If a function’s domain is all real numbers, it is written as \(\mathbb{R}\), and for ordered pairs of real numbers, we use \(\mathbb{R}^2\). So, if a function in two variables like \(f(x, y)\) is expressed to have a domain of all ordered pairs of real numbers, we would use the notation \(\{(x, y) | x \in \mathbb{R}, y \in \mathbb{R}\}\) or simply \(\{(x, y) \in \mathbb{R}^2\}\).

The beauty of set notation lies in its clarity and universality. Instead of writing long, convoluted descriptions of what numbers are included or excluded from a set, set notation lays it all out in a compact, easily understandable form.